We give a proof in the Appendix  for reference. There is a similar proposition for the asymptotic behavior of RLPN decoding used together with [1]} which is given in the Appendix . We have used them for producing the complexity curves given in Figure REF  which display the various complexities of the RLPN decoders we have presented. Even if there is a tiny improvement by using [1]} instead of [3]} the two curves are nearly indistinguishable. A perspective of improvement of our algorithm could be to produce the parity-check equations by using more recent ISD techniques than [1]}, in particular [5]} or [6]} which both use nearest-neighbor search. Our preliminary results using in particular [5]} do not provide significant improvement, we have only been able to achieve a very slightly better complexity for rates close to \(0.2\) .
